<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4/A</documentType>

    <periodOfReport>2013-06-04</periodOfReport>

    <dateOfOriginalSubmission>2013-06-06</dateOfOriginalSubmission>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000070487</issuerCik>
        <issuerName>NATIONAL RESEARCH CORP</issuerName>
        <issuerTradingSymbol>NRCIA/B</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001053133</rptOwnerCik>
            <rptOwnerName>Hays Michael D.</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O NATIONAL RESEARCH CORPORATION</rptOwnerStreet1>
            <rptOwnerStreet2>1245 &quot;Q&quot; STREET</rptOwnerStreet2>
            <rptOwnerCity>LINCOLN</rptOwnerCity>
            <rptOwnerState>NE</rptOwnerState>
            <rptOwnerZipCode>68508</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>CEO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <transactionDate>
                <value>2013-06-05</value>
                <footnoteId id="F2"/>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F3"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>5000000</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>12.87</value>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>5836300</value>
                    <footnoteId id="F4"/>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">On May 22, 2013, the Issuer completed a recapitalization pursuant to which it, among other things, issued a stock dividend of three shares of class A common stock for each share of then-existing common stock and reclassified each share of its then-existing common stock as one-half of one share of class B common stock.  All fractional shares of class B common stock were cashed-out in the recapitalization.</footnote>
        <footnote id="F2">The Form 4 filed on June 6, 2013 reported that 5,000,000 shares were sold on June 4, 2013 at a price of $14.15.  The reporting person actually sold 5,000,000 shares on June 5, 2013 at a price of $12.87.  This Form 4 is being filed to correct the transaction date and sale price.</footnote>
        <footnote id="F3">Shares were sold to The Michael and Karen Hays Grandchildren's Trust Dated March 9, 2009 in return for a promissory note.</footnote>
        <footnote id="F4">Does not include 375,000 shares of class A common stock currently held in a certain GRAT agreement, all or a portion of which will be returned to Mr. Hays over the next two years.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Russell E. Ryba, Attorney-In-Fact for Michael D. Hays</signatureName>
        <signatureDate>2013-06-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
