{"id":2368,"date":"2026-11-09T06:20:13","date_gmt":"2026-11-09T06:20:13","guid":{"rendered":"https:\/\/prwatech.in\/blog\/?p=2368"},"modified":"2026-11-10T16:18:48","modified_gmt":"2026-11-10T16:18:48","slug":"z-score-tutorial","status":"publish","type":"post","link":"https:\/\/prwatech.in\/blog\/data-science\/data-science-modules\/z-score-tutorial\/","title":{"rendered":"Z- Score"},"content":{"rendered":"

Z Score: Definition, Formula and Worked Examples<\/b><\/strong><\/h1>\n

Sarah scores 70 out of 100. Is that good? It depends on the class. If the average was 60, she did well. If the average was 85, she did badly. A raw score means little until you compare it with the group. A z score does exactly that: it tells you how far a value sits from the average, counted in standard deviations.<\/p>\n

It is one of the simplest and most used tools in data science. Analysts use it to compare values measured on different scales, to find outliers, to scale features before a model, and to turn a normal curve question into a table lookup.<\/p>\n

This tutorial explains what a z score is, how to read positive and negative values, the formula, a fully worked example with the answer, how to find percentiles and work backwards, how to spot outliers, and how to calculate z scores in Python.<\/p>\n

Z Score in Data Science: Definition<\/h2>\n

A z score, also called a standard score, shows how many standard deviations a data point lies from the mean of its group.<\/p>\n

How to Read a Z Score?<\/h3>\n
    \n
  • z = 0: the value equals the mean.<\/li>\n
  • z = 1: the value is one standard deviation above the mean.<\/li>\n
  • z = 2: two standard deviations above the mean.<\/li>\n
  • z = 3: three standard deviations above the mean. This is rare.<\/li>\n
  • A positive z: the value is above the mean.<\/li>\n
  • A negative z: the value is below the mean. For example, z = minus 1.5 is one and a half standard deviations below.<\/li>\n<\/ul>\n
    \"z<\/div>\n

    The size of z matters, not just its sign. A z of 2.4 and a z of minus 2.4 are equally far from the mean, on opposite sides.<\/p>\n

    The live page says that a z score comes under probability. It is calculated as part of descriptive statistics, where it is a way of standardizing data. It enters probability when you read the area under the normal curve for a z value.<\/p>\n

    Z Score Formula<\/h2>\n
    \"z<\/div>\n
    z = (x minus mean) divided by standard deviation<\/div>\n
      \n
    • x = the value you are checking.<\/li>\n
    • mean = the average of the group.<\/li>\n
    • standard deviation = the spread of the group.<\/li>\n<\/ul>\n

      In symbols, for a population: z = (x minus mu) divided by sigma. For a sample: z = (x minus x bar) divided by s.<\/p>\n

      Steps to Calculate a Z Score<\/h2>\n
        \n
      1. Find the mean of the data.<\/li>\n
      2. Find the standard deviation.<\/li>\n
      3. Subtract the mean from the value.<\/li>\n
      4. Divide by the standard deviation.<\/li>\n
      5. Read the result: its sign gives the direction and its size gives the distance.<\/li>\n<\/ol>\n

        Z Score Example<\/h2>\n
        \"z<\/div>\n

        Example 1: Sarah’s Score<\/h3>\n

        The live page: Sarah scored 70 out of 100. The class mean was 60 and the standard deviation was 15. How many students scored higher or lower? The page does not answer. Here is the full solution.<\/p>\n

          \n
        • Z = (70 minus 60) divided by 15 = 10 divided by 15 = 0.667.<\/li>\n
        • Meaning: Sarah is about two thirds of a standard deviation above the class mean.<\/li>\n
        • Area to the left of 0.667 = 0.7475 (from a z table or software).<\/li>\n<\/ul>\n

          If the marks follow a normal distribution, about 74.75 percent of the class scored lower than Sarah, and about 25.25 percent scored higher. The normal assumption matters. Without it, the z score still tells you the distance from the mean, but the percentage may be different.<\/p>\n

          Example 2: Working Backwards<\/h3>\n

          You can turn a z score back into a raw score: x = mean plus z times standard deviation.<\/p>\n

            \n
          • A z of 1.5 in Sarah’s class is a score of 60 + 1.5 times 15 = 82.5.<\/li>\n
          • The 90th percentile has a z of 1.282, so the score is 60 + 1.282 times 15 = 79.22. A student needs about 79 marks to beat 90 percent of the class.<\/li>\n<\/ul>\n

            Example 3: Comparing Two Exams<\/h3>\n

            Exam A: the mean is 70, the standard deviation is 8, and a student scores 82. Exam B: the mean is 500, the standard deviation is 100, and a student scores 640.<\/p>\n

            \n\n\n\n\n\n\n
            Z scores for the two exams<\/caption>\n
            Point<\/th>\nExam A<\/th>\nExam B<\/th>\n<\/tr>\n<\/thead>\n
            Score<\/th>\n82<\/td>\n640<\/td>\n<\/tr>\n
            Z score<\/th>\n1.5<\/td>\n1.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

            The 82 is slightly stronger, at 1.5 standard deviations above its mean against 1.4. This is the main use of z scores: putting different scales on one ruler.<\/p>\n

            Example 4: A Small Class<\/h3>\n

            Eight students scored 55, 60, 65, 70, 48, 72, 80 and 58. The mean is 63.5. The population standard deviation is 9.644, and the sample standard deviation is 10.309. Their z scores, using the population standard deviation, are:<\/p>\n

            \n\n\n\n\n\n
            Marks and z scores for eight students<\/caption>\n
            Mark<\/th>\n55<\/th>\n60<\/th>\n65<\/th>\n70<\/th>\n48<\/th>\n72<\/th>\n80<\/th>\n58<\/th>\n<\/tr>\n<\/thead>\n
            Z score<\/th>\nMinus 0.88<\/td>\nMinus 0.36<\/td>\n0.16<\/td>\n0.67<\/td>\nMinus 1.61<\/td>\n0.88<\/td>\n1.71<\/td>\nMinus 0.57<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

            Two things to notice. The z scores are negative for marks below the mean, and the 80 stands out most, at 1.71. And after standardizing, the z scores have a mean of 0 and a standard deviation of 1. Check the code below.<\/p>\n

            Population or sample? If your data are the whole group, divide by the population standard deviation. If they are a sample, the sample version, with n minus 1, is usually used. The z scores change a little. Python’s stats.zscore uses the population version unless you set ddof to 1.<\/p>\n